Informationally complete joint measurements on finite quantum systems

TL;DR

This paper demonstrates the existence and properties of informationally complete joint measurements for conjugated observables in finite quantum systems, highlighting differences between odd and even dimensions.

Contribution

It introduces the concept of informationally complete joint measurements for conjugated observables and characterizes their implementation and uniqueness depending on system dimension.

Findings

Joint measurements can identify all quantum states from outcomes.

Implementation as sequential measurements is possible.

Informational completeness depends on the system's dimension (odd vs. even).

Abstract

We show that there are informationally complete joint measurements of two conjugated observables on a finite quantum system, meaning that they enable to identify all quantum states from their measurement outcome statistics. We further demonstrate that it is possible to implement a joint observable as a sequential measurement. If we require minimal noise in the joint measurement, then the joint observable is unique. If the dimension d is odd, then this observable is informationally complete. But if d is even, then the joint observable is not informationally complete and one has to allow more noise in order to obtain informational completeness.